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Theorem 2eu5 2185
Description: An alternate definition of double existential uniqueness (see 2eu4 2184). A mistake sometimes made in the literature is to use  E! x E! y to mean "exactly one  x and exactly one  y." (For example, see Proposition 7.53 of [TakeutiZaring] p. 53.) It turns out that this is actually a weaker assertion, as can be seen by expanding out the formal definitions. This theorem shows that the erroneous definition can be repaired by conjoining 
A. x E* y ph as an additional condition. The correct definition apparently has never been published. ( E* means "exists at most one.") (Contributed by NM, 26-Oct-2003.)
Assertion
Ref Expression
2eu5  |-  ( ( E! x E! y
ph  /\  A. x E* y ph )  <->  ( E. x E. y ph  /\  E. z E. w A. x A. y ( ph  ->  ( x  =  z  /\  y  =  w ) ) ) )
Distinct variable groups:    x, y,
z, w    ph, z, w
Allowed substitution hints:    ph( x, y)

Proof of Theorem 2eu5
StepHypRef Expression
1 2eu1 2181 . . 3  |-  ( A. x E* y ph  ->  ( E! x E! y
ph 
<->  ( E! x E. y ph  /\  E! y E. x ph )
) )
21pm5.32ri 616 . 2  |-  ( ( E! x E! y
ph  /\  A. x E* y ph )  <->  ( ( E! x E. y ph  /\  E! y E. x ph )  /\  A. x E* y ph ) )
3 eumo 2141 . . . . 5  |-  ( E! y E. x ph  ->  E* y E. x ph )
43adantl 448 . . . 4  |-  ( ( E! x E. y ph  /\  E! y E. x ph )  ->  E* y E. x ph )
5 2moex 2172 . . . 4  |-  ( E* y E. x ph  ->  A. x E* y ph )
64, 5syl 16 . . 3  |-  ( ( E! x E. y ph  /\  E! y E. x ph )  ->  A. x E* y ph )
76pm4.71i 610 . 2  |-  ( ( E! x E. y ph  /\  E! y E. x ph )  <->  ( ( E! x E. y ph  /\  E! y E. x ph )  /\  A. x E* y ph ) )
8 2eu4 2184 . 2  |-  ( ( E! x E. y ph  /\  E! y E. x ph )  <->  ( E. x E. y ph  /\  E. z E. w A. x A. y ( ph  ->  ( x  =  z  /\  y  =  w ) ) ) )
92, 7, 83bitr2i 263 1  |-  ( ( E! x E! y
ph  /\  A. x E* y ph )  <->  ( E. x E. y ph  /\  E. z E. w A. x A. y ( ph  ->  ( x  =  z  /\  y  =  w ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 5    <-> wb 175    /\ wa 357   A.wal 1521   E.wex 1526   E!weu 2102   E*wmo 2103
This theorem was proved from axioms:  ax-1 6  ax-2 7  ax-3 8  ax-mp 9  ax-5 1522  ax-6 1523  ax-7 1524  ax-gen 1525  ax-8 1612  ax-11 1613  ax-17 1617  ax-12o 1653  ax-10 1667  ax-9 1673  ax-4 1681  ax-16 1915
This theorem depends on definitions:  df-bi 176  df-or 358  df-an 359  df-tru 1309  df-ex 1527  df-nf 1529  df-sb 1872  df-eu 2106  df-mo 2107
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